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![Page 1: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/1.jpg)
Introduction to Probability Theory ‧3-1‧
Speaker: Chuang-Chieh LinAdvisor: Professor Maw-Shang Chang
National Chung Cheng University
Dept. CSIE, Computation Theory Laboratory
January 25, 2006
- Preliminaries for Randomized Algorithms
![Page 2: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/2.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 22
Outline
• Chapter 3: Discrete random variables– Bernoulli and binomial distributions
– Geometric distribution
– Negative binomial distribution
![Page 3: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/3.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 33
Bernoulli trials ( 伯努利試驗 )
• A Bernoulli trial is an experiment with two different possible outcomes, labeled successsuccess and failurefailure. The sample space for a single Bernoulli trial is defined as T = {s, f}, where s represents the outcome success and f represents the outcome failure.
![Page 4: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/4.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 44
Bernoulli random variable
• If an experiment consists of a single Bernoulli trial with parameter p (so that P({s}) = p, and we denote q = 1 – p) and we let X be the number of successes to occur, then X is called a Bernoulli random variable with parameter p.
• Its probability function is very simple:
otherwise ,0
1,0for ,)(
1 xqpxp
xx
X
![Page 5: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/5.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 55
Bernoulli random variable (contd.)
• Mean and variance for a Bernoulli random variable X with parameter p:
pqppppXX
XpppX
X
XX
)1(])[E(][E
][E)1(1)0(0][E2222
2
![Page 6: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/6.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 66
• Many experiments can be modeled as a sequence of independent Bernoulli trials.
• For example,– Ten scratch-off lottery tickets are purchased; each ticket
either will or will not win some prize, where p is the probability of a success occurring for each.
– Each of 100 patients with the same affliction is given medication A ; each patient will either be cured or not, with the same success probability p.
![Page 7: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/7.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 77
Binomial random variable( 二項隨機變數 )
• If Y is the number of success to occur in n repeated, independent Bernoulli trials, each with probability of success p, then Y is a binomial random variable with parameter n and p. The range for Y is RY = {0, 1, 2,…, n}, and its probability function is
where q = 1 – p
otherwise. ,0
.for ,)( Y
yny
Y
Ryqpy
nyp
![Page 8: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/8.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 88
• 假設老王買了 10 張刮刮樂彩券。假設每張彩券贏得某個獎項的機會是 1/9 ,而彩券彼此互相獨立。因此每張彩券可視為一次 Bernoulli trial ;若令 X 代表會中獎的彩券張數,則 X 具有 n = 10, p = 1/9 的binomial distribution 。
• 則
• 老王的彩券至少有三張會中獎的機率,便是
.10,,2,1,0 ,9
8
9
110)(
10
xx
xpxx
X
0906.09
8
9
110)()3(
1010
3
10
3
xx
xxX x
xpXP
![Page 9: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/9.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 99
Means and variances for binomial random variables
np
qpnp
qpx
nnp
qpx
nxX
n
n
x
xnx
n
x
xnxX
)(
1
1
][E
1
1
)1()1(1
0
![Page 10: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/10.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1010
Means and variances for binomial random variables (contd.)
•
•
)1()]1([E)]1([E
have we
)]1([E][E)]1([E][E Since
22
2
XXXXX
X
XXXX
XXXXXX
2
22
2
22
20
)1(
)()1(
)!()!2(
)!2()1(
)!(!
!)1()1()]1([E
pnn
qppnn
qpxnx
npnn
qpxnx
nxxqp
x
nxxXX
n
n
x
xnx
n
x
xnxn
x
xnx
![Page 11: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/11.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1111
Means and variances for binomial random variables (contd.)
• Thus
npq
pnp
npnppnn
XX XXX
)1(
)1()1(
)1()]1([E2
2
![Page 12: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/12.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1212
• Before introducing the other probability distribution, we have to be familiar to infinite geometric series first.
![Page 13: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/13.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1313
Infinite geometric series
• When | q | < 1,
1
1
1
1 limlim
1
0 qq
n
n
n
i
i
n
)1(
1
1
12
0 qqdq
dq
dq
d
i
i
)1(
2
)1(
132
0
2
qqdq
dq
dq
d
i
i
![Page 14: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/14.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1414
Infinite geometric series (contd.)
• Then we will obtain that
• An exercise.
10 )1(
1
k
kj
kj
i
i
k
jq
i
ik
![Page 15: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/15.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1515
Geometric distribution ( 幾何分佈 )
• Let N be the trial number of the first successthe first success in a sequence of independent Bernoulli trials, each with parameter p. The probability function for N is
N is called a geometric random variable with parameter p.
otherwise. ,0
},3,2,1{for ,)(
1 Nn
N
Rnpqnp
![Page 16: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/16.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1616
Memoryless property ( 失憶性 )
• If N is a geometric random variable with parameter p, then
where a and b are any positive integers. This is the only discrete probability law to have this memoryless property.
).()|( aNPbNbaNP
![Page 17: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/17.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1717
• 舉例來說:
• 假設我們現在要搜尋一個得 SARS 的病患,而當我們找到第一個病患就停止搜尋。不同的人之間為互相獨立的 Bernoulli trials , p = 0.1 。
• 假設我們已經檢查了 8 個人,都還沒出現成功的試驗 ( 找到一個得SARS 的病患 ) ,則下一個人是 SARS 病患的機率並不會因此改變。這即為失憶性 (memoryless property) 。
![Page 18: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/18.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1818
Means and variances for geometric random variables
•
p
qp
qqp
pqnnpnNn
n
nNN
1
)1(
1
)321(
)(][E
2
2
1
1
1
![Page 19: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/19.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 1919
Means and variances for geometric random variables (contd.)
• Since
We have
23
2
2
1
1
2
)1(
2
)1262(
)1(
)()1()]1([E
p
q
qpq
qqpq
pqnn
npnnNN
n
n
nN
22 1
112
)1()]1([E][Var
p
q
ppp
q
NNN NN
![Page 20: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/20.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 2020
Negative binomial distribution ( 負二項分佈 )
• Independent Bernoulli trials, each with probability of success p, are performed until the rth success occurs. The number of trials required, Nr , is called a negative binomial random variable with parameter r, p; its probability function is as follows:
otherwise. ,0
},2,1,{ ,1
1)(
rrrRnqpr
nnp N
rnr
Nr
![Page 21: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/21.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 2121
Means and variances for negative binomial random variables
•
2
,
p
rq
p
r
r
r
N
N
![Page 22: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/22.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 2222
Means and variances for negative binomial random variables (contd.)
•
•
p
r
qrpq
r
nnpN
rnr
rrnrr
1)1(
1
1
1][E
2
1,1 where,2)1(
2)!(!
)!1(
)!(!
!)21(
)!()!1(
)!1()1()]1([E
p
qprr
rrnnqr
nrpq
r
nprr
qr
nrpq
rnr
nrp
qrnr
nnrp
qprnr
nnnNN
rn
rnr
rn
rnr
rn
rnr
rn
rnr
rn
rnr
rn
rnrrr
![Page 23: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/23.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 2323
Means and variances for negative binomial random variables (contd.)
• Thus
2
22
1
p
rq
p
r
p
r
p
qprr
rN
![Page 24: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/24.jpg)
Thank you.
![Page 25: Introduction to Probability Theory ‧ 3- 1 ‧ Speaker: Chuang-Chieh Lin Advisor: Professor Maw-Shang Chang National Chung Cheng University Dept. CSIE, Computation.](https://reader031.fdocuments.in/reader031/viewer/2022032308/56649f565503460f94c79dea/html5/thumbnails/25.jpg)
Computation Theory Lab., Dept. CSIE, CCU, TaiwanComputation Theory Lab., Dept. CSIE, CCU, Taiwan 2525
References
• [H01] 黃文典教授 , 機率導論講義 , 成大數學系 , 2001.
• [L94] H. J. Larson, Introduction to Probability, Addison-Wesley Advanced Series in Statistics, 1994; 機率學的世界, 鄭惟厚譯, 天下文化出版。
• [M97] Statistics: Concepts and Controversies, David S. Moore, 1997; 統計,讓數字說話, 鄭惟厚譯 , 天下文化出版。
• [MR95] R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, 1995.